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Math Professors Put Their Stamp on Research

Ready for a math quiz? If that's too intimidating, call it a critical thinking question. If you have a roll of 3-cent stamps and one of 7-cent stamps, what is the highest postage amount that is impossible to make with your stamps?

Obviously you could send a 3-cent letter. A 4- or 5-cent letter? Impossible. If you keep going -- a 6-cent letter with two 3-cent stamps, and so on -- you eventually discover that 11 cents is the largest amount that would be impossible (nevermind the miracle of how you might fit hundreds of stamps onto your tiny envelope).

Guess what? Typical of math, there's an equation that could have given you this answer in a snap. Take the product of the two stamp amounts (3 x 7 = 21) and subtract their sum (3 + 7 = 10): 21 - 10 = 11.

Now add a third roll of different-valued stamps to your collection, and it gets more complicated. In fact, a simple equation to find the highest impossible postage amount does not exist -- hence the Frobenius Problem, as it's formally called. And it's this problem that inspired the work of a mathematics team this summer at Willamette, led by assistant professors Colin Starr and Inga Johnson.

Johnson and Starr's specific research question is too complex for an article in anything other than an academic mathematics journal (in fact, they're publishing a paper on it soon in the Journal of Integer Sequences). But what's more important in this case is that the professors are bending students' minds in ways they may not be used to from math class -- moving beyond problem-solving using equations puzzled out by other brilliant mathematicians, and instead conducting research to write their own theorems. "With research, you can't just flip to the back of the book to get your answers. There is no book," Starr says.

Since coming to Willamette several years ago, Johnson and Starr have made it their goal for their students to experience math research. Remember writing proofs in geometry class? You couldn't just solve a problem, but you had to show how you solved it and why the formulas worked the way they did. Same goes for math research. You must prove that your proposed equation will work in every possible instance -- not an easy task.

To share this type of thinking with their students, Johnson and Starr hosted a summer research program for a few years to study questions relating to the Frobenius Problem. "You don't normally get to do research in the classroom as an undergraduate," Johnson says. "So being at the forefront of research is pretty exciting for students."

Then last spring, their project got bigger. Much bigger. They applied for a three-year, half-million-dollar National Science Foundation grant and got it -- on their first try. "We thought we would be turned down because that's what's supposed to happen," Starr says. "We were thrilled," Johnson adds.

With the grant, they created the Willamette Valley Consortium for Mathematics Research, which includes Willamette, Linfield College, the University of Portland and Lewis & Clark College. This summer, teams of four undergraduate students, one high school or community college teacher and two faculty mentors met for eight weeks on each of the campuses to wrestle with various mathematical questions. Those convening at Willamette worked with the Frobenius Problem.

But before the teams could even start working, they had to understand what it means to be a researcher. "It's worse than just not being able to go to the book for the answers," Starr says. "You can't even go to the book for the question."

Starr and Johnson approach math with humor and a sense of ease that could make almost anyone comfortable hearing their theories. They encouraged this math-phobic writer to take one of their Contemporary Mathematics classes -- "You'll be surprised what you'll learn," Starr says. In fact, they urge everyone who says they're "bad at math" -- a commonly uttered phrase -- to re-think the statement.

Math suffers from a negative stereotype that both professors are fighting madly to change. "It's not acceptable for people to say 'I can't read or write,' but many in our society think it's OK to be bad at math," Johnson says. "Math is fun and cool, and those who don't study it are missing out on learning about a beautiful and dynamic subject."